Are All Red Cards Created Equal?

Chris Smalling of Manchester United, right, receives a red card during the Barclays Premier League match between Manchester City and Manchester United on November 2nd.

Photograph by Howard Walker / Anadolu Agency / Getty

It hasn’t been the easiest of seasons for Manchester United, and especially not for Chris Smalling, who, having missed half of his team’s games this season, was sent off in the thirty-ninth minute of last Sunday’s derby against Manchester City. The expulsion of a defender like Smalling can be very costly, especially so early in the match (Manchester United did end up losing). Smalling will also serve a one-game suspension, but it could have been worse; at least the red card didn’t come with a penalty kick.

It’s hard to imagine a bigger blow to a team than the so-called “triple punishment,” which has taken its toll in soccer leagues around the world since a rule change in 1990. When a player fouls an attacker inside the box surrounding his own goal, the referee is supposed to call a penalty and may also show the guilty party a red card. The offending player not only has to leave the game before the penalty kick, which is usually scored against his team, but must also miss his team’s next match.

The idea that one action can hamstring a team in two consecutive games has provoked plenty of debate. Soccer officials have repeatedly suggested that the triple punishment is too harsh for a solitary foul, no matter where it occurs. But this year, when the European soccer supremos at UEFA asked FIFA’s rules board to change the policy, it refused. The board had its reasons; as some writers have pointed out, reducing the suspension to a warning could make fouls committed inside the box less costly than those committed outside. Of course, we could also make the simple argument that the offending player deserves the suspension, and that the victimized team deserves a high-probability chance to score a goal.

Despite a quarter century of hullaballoo, very little in the way of data has been used to inform this debate. So, to find out just how costly the triple punishment might be, I took a look at the past four seasons of the English Premier League. Through a series of regressions and other statistical estimates, I tried to figure out how red cards and penalties were related to final scores. All of the data I used were publicly available.

Controlling for home-field advantage and the quality of the opposition, red cards cost the offending teams about .015 goals per minute left in a game, which was close to earlier estimates by Mark Taylor. In other words, if a red card occurred with forty minutes left in a game, the score at the end of the contest was 0.6 goals worse, on average, for the team playing with ten men.

Giving up a penalty kick was similarly costly, with the penalized team surrendering an average of 0.65 additional goals. We might have expected an even higher figure, since players scored seventy-eight per cent of the penalty kicks in the sample. We might also have thought that the teams that gave up penalties were already likely to be on the losing side, but that was not the case; there was no correlation across games between penalty shots and non-penalty shots or goals. A possible explanation is that penalties were essentially random, and teams conceding goals from penalties performed slightly better for the rest of the game as they struggled to make up the deficit.

But what about when the two events coincided? Suffering a red card and a penalty simultaneously caused no additional disadvantage; the scores came out roughly the same as if the two events had happened separately. Adding the effects together, the best guess for the cost associated with receiving a penalty and a red card with, say, forty minutes left in a game is 1.25 goals—much more than the single goal that might have been scored if the original play had continued.

Already, fouling in the box and risking the triple punishment seems like a poor decision, and we’ve only dealt with two of the punishment’s components. So what happened in the next game?

Unless a team successfully appeals a red card, the expelled player is also prohibited from playing in the following league match. Of course, a manager facing this situation has plenty of time to come up with a new lineup; some managers notoriously never play the same lineup twice in a row, anyway. In any event, we wouldn’t expect it to matter whether the red card accompanied a penalty or not.

Yet, apparently, it does. Neither a penalty nor, perhaps surprisingly, a generic red card has any significant effect on a team’s score in their next game. But teams that concede both together give up half a goal more, on average, in the matches that follow. (For the technically inclined, the probability that the true effect was not zero—often called the “confidence level”—came in at eighty-nine per cent.)

Where could this effect have come from? We might suspect that players receiving red cards for fouls that led to penalty kicks were more likely to be goalkeepers and defenders, who were tougher to replace in the lineup—and we’d be right. Of the thirty-one instances over the four seasons studied here, twenty-seven involved defenders and goalkeepers. So was a red card with a penalty simply a marker for weakening the squad?

As it happens, no. If we split the red cards according to the positions of the players who received them, the expulsion of a defender or a goalkeeper did seem to hurt teams more than losing a midfielder or a forward. The former group’s red cards were associated with losses of 0.024 goals for each minute left in a game; for the latter group, it was 0.009 goals. This may indicate that primarily defensive players were tougher to replace—what team doesn’t have a spare striker on the bench?—or that managers simply preferred not to throw on another defender when one was forced to leave the field. That may not have been a winning strategy.

But if we look at the next game, still controlling for the position of the suspended player, the combined effect of an expulsion and a penalty has almost the same magnitude: a loss of 0.44 goals (at a confidence level of eighty-two percent), just a tad less than the 0.50 goals in the previous estimate. Now, this effect could be nothing more than a statistical artifact; a game-by-game analysis of the thirty-one cases would undoubtedly help to reveal the losses’ true causes. Indeed, a match database provided to me by Jimmy Coverdale of Bloomberg Sports suggests that the coincidence of penalties and red cards varies greatly by league and by season.

It may be worth entertaining one other hypothesis, however. Even among defenders and goalkeepers, the ones who were sent off after conceding penalties weren’t just any players. They tended to be the ones who glued their sides together. What is Manchester City without Vincent Kompany? Or Arsenal without Per Mertesacker? Everton without Tim Howard? Roughly a third were not only essential tactically but also captains or vice-captains of their squads. In England’s stolid soccer culture, these players may have felt that it fell to them to take one for the team. But if they had their teams’ best interests in mind they probably shouldn’t have.

Dan Altman is an economist and the founder of North Yard Analytics LLC, a sports-data consultancy.